Insomnia: van Aubel's Theorem

I woke up in the middle of the night and stumbled upon this post in mathstodon about a simple theorem in Euclidean geometry. So I thought it was a good time to dust off my Geogebra app.

Van Aubel's theorem states that if you start with a convex quadrilateral and construct a square on each side, externally to the quadrilateral, then the line segments connecting the centers of opposite squares will be equal in length and perpendicular to each other.

The following images provide a synthetic proof of this theorem. The line segments mentioned are colored in green and red.

The green and red circumference cross on point $O_1$, which is the center of the blue circumference. By construction, the blue circumference contains points $M_1$ and $K_1$. Hence the red and green segments are of equal length.

The purple line is the perpendiculat to the green line that passes through $P_1$. As can be seen, the purple line contains the red segment. Hence the green and red segments are perpendicular.

This concludes the proof of this theorem.

I much clearer proof is following diagram.The red circle around $M$ contains the centers $I$ and $K$ on its diameter as well as point $N$, where both (green) segments joining opposite centers cross, thus forming an inscribed triangle, which proves the segments being perpendicular. 

Finally, each segment can pivot around $I$ and $K$, respectively, meeting at a common point $O$, around which they can pivot again drawing the large red circle that contains $I$ and $K$. This proves both segments are the same length.

The last diagram shows that, in essence, Aubel's theorem hinges on a similar property for triangles. Here $H$ is the midpoint of a side. The yellow circle, centered on the midpoint between $F$ and $G$, shows that the two green segments are perpendicular, while the purple circle, centered on $M$,  proves they are the same length.